V

Plot3D[(E^((x^2) - (y^2)))  , {x, -2, 2}, {y, -2, 2}]

Vx  =  D[V, x]

Vy  =  D[V, y]

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[0]==1,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 70}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 1}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 2}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 4}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 6}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 7}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 8}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 9}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 10}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 11}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 12}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 13}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 20}, \ 
      PlotRange -> All]\)

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[0]==0.1,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 70}]

sol=NDSolve[{x''[t]==- x[t] ,y''[t]== - y[t],x'[0]==1,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 70}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 11}, \ 
      PlotRange -> All]\)

H = {1/2} (x'[t]^2  - y'[t]^2)  +   
          (E^(x[t]^2 - y[t]^2)) /. sol

t = 0

0t = 2

H = {1/2} (x'[t]^2  - y'[t]^2)  +   
          (E^(x[t]^2 - y[t]^2)) /. sol

t = 20

H = {1/2} (x'[t]^2  - y'[t]^2)  +   
          (E^(x[t]^2 - y[t]^2)) /. sol

t =.

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 + y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 + y[t]^2)) y[t],x'[0]==0.1,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 70}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 11}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 20}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 36}, \ 
      PlotRange -> All]\)

Plot3D[(E^((x^2) - (y^2)))  , {x, -4, 4}, {y, -4, 4}]

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[0]==0.1,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 70}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 20}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 50}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 70}, \ 
      PlotRange -> All]\)

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[0]==0.1,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 200}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 81}, \ 
      PlotRange -> All]\)

x'[81] /. sol

y'[81] /. sol

x[81] /. sol

y[81] /. sol

{-3.5393219622404004`}

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[81]== 1.99759047163824234`,y'[81]== 2.07809318835298828`,x[81] == -3.45034135963493016` ,y[81] ==  -3.5393219622404004`},{x, y},{t, 81, 200}]

t =.

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 81, 100}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 81, 120}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 81, 154}, \ 
      PlotRange -> All]\)

H = {1/2} (x'[t]^2  - y'[t]^2)  +   
          (E^(x[t]^2 - y[t]^2)) /. sol

t = 81

t = 120

H = {1/2} (x'[t]^2  - y'[t]^2)  +   
          (E^(x[t]^2 - y[t]^2)) /. sol

t = 154

H = {1/2} (x'[t]^2  - y'[t]^2)  +   
          (E^(x[t]^2 - y[t]^2)) /. sol

t =.

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[0]==0.1,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 200}]

t = 0

H = {1/2} (x'[t]^2  - y'[t]^2)  +   
          (E^(x[t]^2 - y[t]^2)) /. sol

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[0]==0.001,y'[0]==0,x[0] == 0,y[0] == 1},{x, y},{t, 200}]

t =.

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 10}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 40}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 86}, \ 
      PlotRange -> All]\)

x'[86] /. sol

\(\(\(y'\)[86] /. sol
\)\)

x[86] /. sol

y[86] /. sol

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[86]== -0.0185725643144353682`,y'[86]==-0.794088190746185773` ,x[86] ==  -0.0131884257581349695`,y[86] ==  -0.617745581369240603`},{x, y},{t, 86, 300}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 86, 120}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 86, 140}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 86, 159}, \ 
      PlotRange -> All]\)

t =.

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[0]==0.001,y'[0]==0.001,x[0] == 0.01,y[0] == 0.01},{x, y},{t, 200}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 2}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 0.02}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 200}, \ 
      PlotRange -> All]\)

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[0]==0.001,y'[0]==0,x[0] == 0,y[0] == 0},{x, y},{t, 200}]

sol=NDSolve[{x''[t]==-2 (E^(x[t]^2 - y[t]^2)) x[t] ,y''[t]== -2 (E^(x[t]^2 - y[t]^2)) y[t],x'[0]==0.001,y'[0]==0,x[0] == 0,y[0] == 0.001},{x, y},{t, 200}]

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 20}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 5}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 40}, \ 
      PlotRange -> All]\)

\(ParametricPlot[Evaluate[{x[t], y[t]}\  /. \ sol], 
	{t, 0, 200}, \ 
      PlotRange -> All]\)